
(Using the Forbidden Moves)
This is a demonstration of why the two "forbidden moves", two
Reidemeisterlike moves on virtual knot diagrams, are not allowable as
virtual moves. In particular, we demonstrate that allowing the moves allows
us to unknot the trefoil, showing that allowing these moves would render
the theory nonviable as a generalization of ordinary knot theory.
Here we start with a diagram of the trefoil knot which looks like a
figureeight knot with one crossing switched:
Next, we apply a type VII move, then our first instance of a forbidden
move (moving a strand with two real crossings past a virtual
crossing, marked in red):

Next, we do another type VII move, followed by another forbidden move:
 

Next, another type VII followed this time by a type V move:
 

Now we can do a type VII to get rid of two virtual crossings, then
do another type V move:
 

Next, we get rid of another two virtual crossings with a type VII move,
followed by another type VII:
 

Next, we do a another type VII followed by another forbidden move:
 
Now we remove a real crossing with a type I move, then do another
type V move to get us into position for our last application of
a forbidden move:
Now our last forbidden move and the knot is actually unknotted, i.e.
equivalent by legitimate moves to a simple uknotted circle with no
crossings, real or virtual. Getting to the trivial diagram of the
knot, however, will still take us a few steps, so we'll abbreviate
in a couple of instances...
Here we do two type VII moves and a type V:
Now we get rid of some more virtual crossings by two applications of
type VII moves, then two type VI moves:
 <VII x 2> 
 <VI x 2> 

Getting closer, we use a type I move to get rid of another real
crossing, then a type VII:
Finally, a single type I move gets us to the trivial diagram of
the unknot:
 <I> 

Whew! That was a fair amount of work. Fortunately, it's much simpler
to do in terms of Gauss Diagrams:
 <Fh x 2> 
 <Ft x 2>
  <I x 4>
 
Ordinary experience tells us that the trefoil is not the same as the
unknot, and for virtual knot theory to be a true extension of knot theory
it cannot allow the forbidden moves, since to do so would mean that
as a virtual knot, the trefoil would be trivial and the theory using the
forbidden move would not be a generalization of knot theory, but a different
(and not very interesting) theory. Fortunately, virtual knot theory can and
does make sense without allowing the forbidden moves.
Copyright © 2000 Sam Nelson

